Euler's equations for free rotation of a rigid body are given by
. For a circular disk of radius
and negligible thickness, the principal moments of inertia are
. The third Euler equation reduces to
. This is the angular velocity of the plate about its axis of symmetry, and is taken as one of the initial conditions. The first two Euler equations are then satisfied by
. The motion of the rigid body in the space-fixed frame can be expressed in terms of the Euler angles
, using the relations
. If the axis of the plate is initially inclined by an angle
from the vertical, with an angular velocity
, then the motion of the plate is given by
. The rotation of the plate is represented by
, while its precession or "wobbling" is given by
approaches 0, the wobbling frequency approaches twice the rotation frequency, but the wobbling amplitude also decreases.
H. Goldstein, Classical Mechanics
, Cambridge, MA: Addison-Wesley, 1953.